Results from [1]} imply that \(\Psi \)  has a normally hyperbolic invariant manifold \(\widetilde{\Lambda }\)  that is \(O \left(\epsilon ^L \right)\)  close in the \(C^4\)  topology to \(\Lambda \times \Sigma \)  where \(\Lambda \subset M\)  is the normally hyperbolic invariant manifold of \(\widetilde{G} ( \cdot ; \xi )\) . Moreover there is an open set \(\widetilde{U} \subset \widetilde{\Lambda }\)  and a scattering map \(\widetilde{S} : \widetilde{U} \rightarrow \widetilde{\Lambda }\)  such that the \(z\) -component of \(\widetilde{S}(z, \theta , \xi )\)  is \(O \left( \epsilon ^L \right)\)  close in the \(C^3\)  topology to \(S \left( z ; \xi \right)\)  where \(S \left( \cdot ; \xi \right) : U \rightarrow \Lambda \)  is the scattering map corresponding to \(\widetilde{G} ( \cdot ; \xi )\) .
